Effective dynamics of the Schwarzschild black hole interior with inverse triad corrections Hugo A. Morales-Técotl,1, 2, ∗ Saeed Rastgoo,3, 1, 4, y and Juan C. Ruelas1, z 1Departamento de Física, Universidad Autónoma Metropolitana - Iztapalapa San Rafael Atlixco 186, Ciudad de Mexico 09340, Mexico 2Departamento de Física, Escuela Superior de Física y Matemáticas del Instituto Politécnico Nacional Unidad Adolfo López Mateos, Edificio 9, 07738 Ciudad de México, Mexico 3School of Sciences and Engineering Monterrey Institute of Technology (ITESM), Campus León Av. Eugenio Garza Sada, León, Guanajuato 37190, Mexico 4Department of Physics and Astronomy, York University 4700 Keele Street, Toronto, Ontario M3J 1P3, Canada (Dated: January 22, 2021) We reconsider the study of the interior of the Schwarzschild black hole now in- cluding inverse triad quantum corrections within loop quantization. We derive these corrections and show that they are are related to two parameters δb; δc associated to the minimum length in the radial and angular directions, that enter Thiemann’s trick for quantum inverse triads. Introduction of such corrections may lead to non- invariance of physical results under rescaling of the fiducial volume needed to compute the dynamics, due to noncompact topology of the model. So, we put forward two prescriptions to resolve this issue. These prescriptions amount to interchange δb; δc in classical computations in Thiemann’s trick. By implementing the inverse triad cor- rections we found, previous results such as singularity resolution and black-to-white hole bounce hold with different values for the minimum radius-at-bounce, and the mass of the white hole. I. INTRODUCTION As one of the most fascinating predictions of general relativity, black holes have been arXiv:1806.05795v3 [gr-qc] 21 Jan 2021 the subject of much analysis and explorations. Particularly their interior, and the singu- larity located there, has been studied in classical, quantum and semiclassical regimes. The mainstream hope is that the classical singularity will be resolved and replaced by a quan- tum region. However, there are still many open issues to be answered in a satisfactory way. Within loop quantum gravity (LQG) [1, 2], there have been numerous works about quantum black holes and their singularity resolution in both mini- and midi-superspace models, to mention a few [3–15]. One of the most studied models in this context is the Schwarzschild black hole which interior corresponds to a Kantowski-Sachs model, a system with finite de- grees of freedom, and hence a mini-superspace, with a singularity at the heart of it [4, 16]. One of the approaches to quantize this model, inspired by LQG, is polymer quantization ∗ [email protected] y [email protected] z [email protected] 2 [17–20], a technique also used in loop quantum cosmology (LQC) [21–23]. In this quanti- zation the classical canonical algebra is represented in a way that is unitarily inequivalent to the usual Schrödinger representation even at the kinematical level. The root of this inequivalency is the choice of topology and the form of the inner product of this represen- tation, which renders some of the operators discontinuous in their parameters, resulting in the representation not being weakly continuous. On the other hand, unitary equivalency of a representation to the Schrödinger one is guaranteed by the Stone-von Neumann theorem iff all of its premises, including weak continuity of the representation, are satisfied, and the polymer representation does not. This inequivalency translates into new results that are different from the usual quantization of the system, one of them being the resolution of the singularity of the Kantowski-Sachs model. These results however, are accompanied by some issues that we briefly discuss in what follows. In one of the earliest attempts in this approach [3], the authors showed that the singularity can be avoided in the quantum regime, but one of the important issues was the dependence of results on auxiliary parameters that define the size of the fiducial cell. The introduction 2 of this fiducial cell, in this case a cylindrical one with topology I ×S and volume V0 = a0L0, 2 where a0 is the area of the 2-sphere S and L0 is the cylinder’s height, is necessary to avoid the divergence of some of the spatial integrals in homogenous models with some non-compact directions. Particularly it is important to be able to define the symplectic structure. Given that the physical results should not depend on these auxiliary parameters, a new proposal, motivated by the “improved quantization” in LQC [24], was put forward that avoided this dependence and yielded bounded expansion and shear scalars [4]. However, this method also leads to some undesired modified behavior at the horizon due to quantum gravitational effects in vacuum, that are manifestation of the coordinate singularity there. There are also some other recent works that take a bit of a different approach to the problem by looking for an effective metric [25, 26]. 2 In [6], a key modification to the quantization was proposed by choosing to fix a0 := 4πr0 by a physical scale r0. This physical r0 permits one to define a Hamiltonian formulation, and in this way is different in nature from the auxiliary scale L0, which is needed to fix the fiducial cell size to be able to define the symplectic structure. Thus while r0 will be present in physical results in both the classical and quantum theories, these theories should be independent of L0. The proposal in [6], leads to results that are independent of the auxiliary parameters, and while the theory predicts that the singularity is resolved in the quantum gravity regime, no large quantum gravitational effects appear at low curvatures near the horizon as it should be the case. It is worth noting that the anisotropic models suffered from an issue: since these models resolve the singularity, they predicted a “bounce” from a black hole to a white hole, with the mass MW of the resultant white hole not matching with that of the original black hole 4 MB 6= MW , but rather MW / MB. Recent work presented some proposals to deal with it [27] and a different approach was developed in [28, 29] by encompassing the interior region containing the classical singularity with the exterior asymptotic one, which, in the large mass limit, makes the masses of the white and black holes take the same value. See also [30–32] for a different perspective. All previous works on black holes ignore inverse triad corrections, to simplify the problem. However, they are important especially at highly quantum regimes and a more complete quantum gravitational analysis of this model should take them into account. In this work we first use a path integral in phase space including inverse triad quantum corrections. 3 These corrections are known to produce severe issues in non compact cosmological models, among them the dependence of the physical quantities on the auxiliary parameters or their rescaling. We put forward two proposals that, in the case of Schwarzschild black hole, yield a physical description with no reference to fiducial parameters. Finally, we study how these proposals modify the “minimum radius at the bounce”. The structure of this paper is as follows: In section II, we present the background, the relation between the Schwarzschild interior and the Kantowski-Sachs model, and their clas- sical Hamiltonian analysis. In section III, we briefly review how the quantum Hamiltonian constraint is defined. In section IV, the path integral analysis is presented and it is shown how a systematic effective Hamiltonian constraint can be derived from the quantum Hamil- tonian, including inverse triad corrections. Section V is dedicated to presenting some of the important issues that are raised by the presence of the new corrections, and recognizing the root of these issues. In section VI, we present two proposals to deal with the aforementioned issues, and also show their effect upon some physical quantities. Finally, in section VII, we conclude the paper by presenting a summary and a discussion about the results. II. BACKGROUND AND THE CLASSICAL THEORY For Schwarzschild black hole the spacetime metric 2GM 2GM −1 ds2 = − 1 − dt2 + 1 − dr2 + r2 dθ2 + sin2 θdφ2 (2.1) r r where M is the mass of the black hole, the timelike and spacelike curves switch their causal nature into each other for observers that cross the event horizon. Hence the metric of the interior region is obtained by r $ t, 2GM −1 2GM ds2 = − − 1 dt2 + − 1 dr2 + t2 dθ2 + sin2 θdφ2 ; (2.2) t t with t 2 (0; 2GM) and r 2 (−∞; 1). This metric is a special case of a Kantowski-Sachs cosmological spacetime that is given by the metric ds2 = −dτ 2 + A2(τ)dr2 + B2(τ) dθ2 + sin2 θdφ2 : (2.3) The coordinates in which (2.3) is written are Gaussian normal coordinates adapted to the comoving observers, i.e., the time coordinate curves are the worldlines of the free falling objects (e.g. stars) that are at rest with respect to such observers, and are parametrized by their proper time τ. The metric (2.2) can be seen to be derived from (2.3) by the transformation 2GM −1 dτ 2 = − 1 dt2: (2.4) t Choosing positive root of the above, we get ! t − GM GMπ τ = −pt(2GM − t) − GM tan−1 + ; (2.5) pt(2GM − t) 2 4 where the last term in the right hand side is the integration constant and it is chosen such that τ ! 0 for t ! 0 (at singularity), and τ ! GMπ for t ! 2GM (at the horizon), hence τ 2 (0; GMπ). Then τ is a monotonic function of t.